. 2022 Oct 25;51(1):168–192. doi: 10.1080/02664763.2022.2138838

Table 5.

The average estimated regression coefficients, the percentage of the estimated regression coefficients greater than zero, and the percentage of the estimated regression coefficients are significantly different from zero at $5 %$ level of significance, over 500 datasets simulated from the LFR2 method, with the second stage model being a normal linear regression model.

VIF	33	29	25	21	17	15	13	12	11
`Naive Model`
${\hat{β}}_{1}^{n a i v e}$	$-$ 0.172	$-$ 0.215	$-$ 0.240	$-$ 0.270	$-$ 0.313	$-$ 0.315	$-$ 0.332	$-$ 0.312	$-$ 0.328
${\hat{β}}_{2}^{n a i v e}$	$-$ 0.180	$-$ 0.135	$-$ 0.119	$-$ 0.094	$-$ 0.081	$-$ 0.063	$-$ 0.058	$-$ 0.061	$-$ 0.053
${\hat{β}}_{3}^{n a i v e}$	0.859	0.858	0.875	0.891	0.930	0.922	0.947	0.941	0.952
$P ({\hat{β}}_{1}^{n a i v e} > 0)$	0.324	0.242	0.190	0.150	0.098	0.084	0.052	0.054	0.042
$P ({\hat{β}}_{2}^{n a i v e} > 0)$	0.294	0.310	0.326	0.330	0.346	0.392	0.352	0.324	0.346
$P ({\hat{β}}_{3}^{n a i v e} > 0)$	0.992	1.000	0.998	1.000	1.000	1.000	1.000	1.000	1.000
$P (p v ({\hat{β}}_{1}^{n a i v e}) < 5 %)$	0.078	0.104	0.136	0.178	0.246	0.274	0.358	0.336	0.424
$P (p v ({\hat{β}}_{2}^{n a i v e}) < 5 %)$	0.100	0.074	0.070	0.064	0.068	0.082	0.052	0.074	0.060
$P (p v ({\hat{β}}_{3}^{n a i v e}) < 5 %)$	0.736	0.864	0.904	0.986	0.996	1.000	1.000	1.000	1.000
`Principal Component Regression`
`PCR1`
${\hat{β}}^{P C R 1}$	$-$ 0.018	$-$ 0.011	$-$ 0.014	$-$ 0.002	$-$ 0.008	$-$ 0.034	0.013	$-$ 0.024	$-$ 0.019
$P ({\hat{β}}^{P C R 1} > 0)$	0.472	0.482	0.480	0.496	0.488	0.446	0.520	0.464	0.470
$P (p v ({\hat{β}}^{P C R 1}) < 5 %)$	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
`PCR2`
${\hat{β}}_{1}^{P C R 2}$	$-$ 0.018	$-$ 0.011	$-$ 0.014	$-$ 0.002	$-$ 0.008	$-$ 0.034	0.013	$-$ 0.024	$-$ 0.019
${\hat{β}}_{1}^{P C R 2}$	$-$ 0.040	$-$ 0.032	$-$ 0.022	0.022	$-$ 0.066	0.003	$-$ 0.022	$-$ 0.017	0.001
$P ({\hat{β}}_{1}^{P C R 2} > 0)$	0.472	0.482	0.480	0.496	0.488	0.446	0.520	0.464	0.470
$P ({\hat{β}}_{1}^{P C R 2} > 0)$	0.482	0.468	0.482	0.510	0.448	0.500	0.486	0.492	0.508
$P (p v ({\hat{β}}_{1}^{P C R 2}) < 5 %)$	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
$P (p v ({\hat{β}}_{2}^{P C R 2}) < 5 %)$	0.342	0.520	0.634	0.804	0.916	0.940	0.968	0.978	0.994
`Latent Factor Regression`
`LFR1`
${\hat{β}}^{L F R 1}$	0.505	0.506	0.509	0.513	0.513	0.512	0.519	0.518	0.516
$P ({\hat{β}}^{L F R 1} > 0)$	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
$P (p v ({\hat{β}}^{L F R 1}) < 5 %)$	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
`LFR2`
${\hat{β}}_{1}^{L F R 2}$	0.501	0.493	0.500	0.500	0.501	0.500	0.503	0.507	0.501
${\hat{β}}_{2}^{L F R 2}$	1.033	1.000	0.996	0.984	1.009	0.987	1.005	0.988	0.994
$P ({\hat{β}}_{1}^{L F R 2} > 0)$	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
$P ({\hat{β}}_{2}^{L F R 2} > 0)$	0.984	0.994	0.994	1.000	1.000	1.000	1.000	1.000	1.000
$P (p v ({\hat{β}}_{1}^{L F R 2}) < 5 %)$	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
$P (p v ({\hat{β}}_{2}^{L F R 2}) < 5 %)$	0.526	0.728	0.824	0.952	0.984	0.996	1.000	1.000	1.000